cp's OEIS Frontend

A099891 XOR difference triangle of A003188 (Gray code numbers), read by rows.

%I A099891 #17 Jul 02 2020 05:18:43
%S A099891 0,1,1,3,2,3,2,1,3,0,6,4,5,6,6,7,1,5,0,6,0,5,2,3,6,6,0,0,4,1,3,0,6,0,
%T A099891 0,0,12,8,9,10,10,12,12,12,12,13,1,9,0,10,0,12,0,12,0,15,2,3,10,10,0,
%U A099891 0,12,12,0,0,14,1,3,0,10,0,0,0,12,0,0,0,10,4,5,6,6,12,12,12,12,0,0,0,0,11,1
%N A099891 XOR difference triangle of A003188 (Gray code numbers), read by rows.
%C A099891 Main diagonal is A099892, the XOR BINOMIAL transform of A003188. See A099884 for the definitions of the XOR BINOMIAL transform and the XOR difference triangle.
%F A099891 T(n, k) = SumXOR_{i=0..k} (C(k, i)mod 2)*(A003188(n-i)), where SumXOR is the analog of summation under the binary XOR operation and C(k, i)mod 2 = A047999(k, i). T(2^n, 2^n) = 3*2^(n-1) for n>0, with T(1, 1)=1 and T(k, k)=0 elsewhere.
%F A099891 T(n,1) = A006519(n), the lowest 1-bit of n (see formula by Franklin T. Adams-Watters in A003188). - _Kevin Ryde_, Jul 02 2020
%e A099891 Rows begin:
%e A099891 [0],
%e A099891 [1,1],
%e A099891 [3,2,3],
%e A099891 [2,1,3,0],
%e A099891 [6,4,5,6,6],
%e A099891 [7,1,5,0,6,0],
%e A099891 [5,2,3,6,6,0,0],
%e A099891 [4,1,3,0,6,0,0,0],
%e A099891 [12,8,9,10,10,12,12,12,12],
%e A099891 ...
%e A099891 where A003188 fills the leftmost column.
%o A099891 (PARI) {T(n,k)=local(B);B=0;for(i=0,k,B=bitxor(B,binomial(k,i)%2*(bitxor((n-i),(n-i)\2))));B}
%Y A099891 Cf. A047999, A003188 (column k=0), A006519 (column k=1), A099892 (diagonal n=k).
%Y A099891 Other triangles: A099884, A099889, A099893.
%K A099891 nonn,tabl
%O A099891 0,4
%A A099891 _Paul D. Hanna_, Oct 29 2004